Determine whether the series is convergent or divergent. If it is convergent, find its sum. ∑n=1

step 1 to numeralad. My name is Kevin Sharak. Let's look at the infinite series from n equals 1 to infi

Determine whether the series is convergent or divergent. If...

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms, usually expressed in summation notation. The convergence or divergence of an infinite series is determined by the behavior of its partial sums as the number of terms grows without bound.

Convergence and Divergence

Determining the convergence or divergence of a series involves analyzing whether the sequence of its partial sums approaches a finite limit. Convergence means that the series sums to a specific value, while divergence indicates that the series does not settle at a finite result.

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a fixed common ratio. Geometric series have a well-known convergence criterion: they converge if the absolute value of the common ratio is less than one, in which case the sum can be calculated using the formula S = a/(1 - r), where a is the first term and r is the common ratio.

Series Decomposition

Series decomposition is a technique where a complex series is split into simpler component series that can be analyzed individually. This is often useful when each component can be recognized as a standard series type, such as a geometric series, allowing for easier determination of convergence and computation of sums.

Transcript

0:00 To numeralad.

00:01 My name is kevin sharak.

00:02 Let's look at the infinite series from n equals 1 to infinity, defined by 1 plus 2 to the n over 3 to the n.

00:19 And we're curious whether or not this series converges.

00:23 Well, one thing we could look at is we could do the nth term test.

00:27 We could run a limit test on this, and i encourage you to do so.

00:31 So we do as n approaches infinity of this inside term here.

00:35 But before even getting too far into this you should see that this this number here is going to follow a similar pattern to the problem that we just did before so let me show you what i mean this can be rewritten and regrouped so we can have limit as n goes to infinity and we could write this separately we could have one over three to the n plus two thirds to the end good well what's going to happen this is actually going to drop off to zero, as is this one.

01:15 So in fact, as n goes off to infinity, this is going to be zero.

01:20 Okay, so that means not that we necessarily converge, but that we don't diverge.

01:28 Well, let's continue with a couple other ways that we can manipulate this expression then and break it down.

01:33 Perhaps we could write this as the summation from n equals 1 to infinity of 1 over 3 to the n plus 2 the n over 3 to the end.

01:47 And we'll keep this.

01:49 That's this separated.

01:50 What would be the advantage? well, the advantage would be that i could break the summation into two.

01:54 Remember that this summation notation is nothing more than a compressed version of addition, and there's no reason why i can't separate addition into two different sets of addition.

02:04 Whether it's an infinite amount of addition or not, i have the ability to perform certain parts of the addition first at no consequence.

02:14 Well, you might notice that i'm writing this second term to look very much like a geometric series.

02:20 So let's see...

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